Recommended economics writing: Link Exchange

1. A Unifying Principle for Economics (Unlearning Economics)
2. Krugman doesn’t understand IS-LM (Steve Keen)
3. Sraffa and the Marshallian System (Naked Keynesianism)

Is the Sraffian Critique of the Labor Theory of Value Correct?

1. Introduction. I've been reading (and re-reading) Steve Keen's Debunking Economics for a while now. He summarizes the Sraffian critique of the Labor theory of value (or its Marxist variant; I'll use the acronym "LTV" for short) using the following system:

28 units iron + 56 units labor → 56 units iron
16 units iron + 16 units labor → 48 units gold
12 units iron + 8 units labor → 8 units corn

It is assumed that it takes 5 units corn to sustain labor.

If we use the LTV, then we set the value created by 1 unit of labor to unity. Thus we have the system become

28 v_i + 56 = 56 v_i
16 v_i + 16 = 48 v_g
12 v_i + 8 = 8 v_c

where v_i is the value of 1 unit iron, v_g the value for 1 unit gold, and v_c the value for 1 unit corn.

Objection. This seems a bit too static...I'm not going to invoke particular interpretations of the LTV (*cough*TSSI*cough*), but it seems like we are mixing apples and oranges when we assume the value of the iron input is the same as the value of the iron output.

2. Determining Values. We can solve this system of equations to find that v_i = 2, v_g = 1, and v_c = 5.

In a more useful, although slightly misleading, form we write: 4 units gold = 2 units iron = 1 unit corn.

Sraffians leap to the conclusion "This isn't what Sraffa's model gives us!"

I would like to ask the simple question: "Wait, let us see: does this work?"

3. Reproduction. Let us start with the iron sector. It keeps 28 units for its own reproduction.

The iron sector sells 12 units to the corn sector. For how much? Well, 12 units of corn = 6 units corn.

The iron sector sells 16 units to the gold sector, for 32 units gold (wow!).

To finish this up, the gold sector needs (¹⁄₅) × 5 units corn, i.e., 1 unit corn. So it buys 1 unit corn from the corn sector for 4 units gold.

The corn sector keeps 1 unit of corn, and has 4 units gold.

We see that reproduction can occur.

4. Profits. The only "problem" I see: we don't have a "uniform rate of profits". What are the profits?

The iron sector has its profits consist of 2.5 units corn + 32 units gold.

The gold sector has its profits consist of nothing: it has no profits.

The corn sector has its profits be 0.5 units corn + 4 units gold.

I suppose, in closing, I have a controversial question (for both Marxists and non-Marxists): is there any reason we should expect a uniform rate of profit across all industries?

Addendum (February 5, 2014 at 7:05AM [PST]). It dawns on me in classical economics, the following conditions are equivalent:

1. The economy is in equilibrium
2. The rate of profits is uniform across all sectors
3. Commodities are exchanged at prices determined by their value (there is no markup, it's just the value expressed in units of money)

I suppose one could take this result and argue "Aha, this must mean Marx works in disequilibrium!" But the problem one must ask is: why would commodities be traded at their value? Perhaps this is the "proof by contradiction" Sraffians are going for...

References.

1. Steve Keen, Debunking Economics. First ed.,
Zed Press, 2001.
2. Ian Steedman, Marx after Sraffa. New Left Books, 1977.

Notes on Sraffa's Production, Chapter 5

36. Introductory

• Sraffa will prove there exists precisely one way to transform a given economic system into a Standard system.

37. Transformation into a Standard system always possible

• We may show this using an "imaginary experiment" (Sraffa's term for a "thought experiment"?).
• It involves two types of alternating steps:
  1. Changing the proportions of the industries;
  2. Reducing the same ratio the quantities produced by all industries, while leaving unchanged the quantities used as means of production.
• We adjust the proportions of all industries in our system such that: every basic commodity produces more than strictly necessary for replacement. (I.e., every basic commodity has surplus.)
  • Is this always possible mathematically?
• Suppose we gradually reduce the product of all industries, slowly and successively in small steps…but without interfering with the quantities of labor and means of production they employ.
  • I honestly don't see how this is done. We decrease the output without changing the input?
• When the cuts reduce production for any one commodity to the minimum level required for replacement, we re-adjust the proportions of the industries so there's a surplus for each product again (while keeping constant the quantity of labor employed in the aggregate.)

  We can always do this, provided there exists a surplus for some commodities and deficit of none

• We continue alternating between cuts and re-establishing a surplus for each product until we reach the point where products have been reduced such that all-round replacement is just possible without leaving anything as surplus.
  • Remark. Notice this is an algorithm! Sraffa's mathematical approach may be described as algorithmic or constructive (the latter for mathematicians, the former for everyone else). Or, at least, this is what K Velupillai has pointed out (pdf)...
• With increasing the quantity produced in each sector by a uniform rate, we are able to restore the original conditions. We do not disturb the proportions to which the industries have been brought. The uniform rate restoring the original conditions of production is R, and the proportions attained by the industries are the Standard proportions!

38. Why question of uniqueness arises

• Is the Standard system unique, or are there other ways to get the same result?
• The equations of the q-system §33 are reducible to an equation of the k-th degree in R. The fundamental theorem of algebra tells us there are at most k different solutions!
• It is sufficient to prove there cannot be more than one value of R which corresponds to an all-positive set of q's. This implies uniqueness of the Standard System.

39. Prices Positive at all wage levels

• First, we must show – as there always exists a possible set of multipliers (§37) – there exists at all values of wage (including zero) a set of prices satisfying the condition of replacement of the means of production with uniform profits. I.e., there exists a set of positive values of p's.
• We consider w = 1 where, since prices equal labor costs (§14), the values of the p's must necessarily all be positive.
  • If the value of w is moved continuously from 1 to 0, the values of the p's will also move continuously…so any p that becomes negative must pass through 0.
  • However, while wages and profits are positive, the price of no commodity can become 0 until the price of at least one of the other commodities entering its means of production becomes negative.
  • Thus, since no p can become negative before any other, none can become negative at all.
  • Footnote. For this proof to be complete, we must show that the p's representing prices of basic products cannot become negative through becoming infinite—unlike the p's of non-basics which can do so. Sraffa shows this in "Note on Self-reproducing Non-basics" (Appendix B).

40. Production equations with zero wages

• For comparison purposes, we rewrite here the production equations as they appear when wages vanish (i.e., when w = 0).
• The labor terms may be omitted (since we multiply them with 0), and we use the maximum rate of profits R for r.
• We can take the price of any one of the commodities as unity.
• The production system becomes

  (A_ap_a + B_ap_b + … + K_ap_k)(1 + R) = Ap_a
  (A_bp_a + B_bp_b + … + K_bp_k)(1 + R) = Bp_b
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  (A_kp_a + B_kp_b + … + K_kp_k)(1 + R) = Kp_k
  

41. Unique set of positive multipliers

We can show now there can be no more than one set of positive multipliers! We will enumerate the steps in Sraffa's proof…

1. Let R' be a possible value of R to which there correspond positive prices p'_a, p'_b, …, p'_k and positive multipliers q'_a, q'_b, …, q'_k.

   Let R" be another possible value of R with corresponding prices p"_a, …, p"_k and multipliers q"_a, …, q"_k.

   We must prove it is impossible for the q"'s to all be positive.

2. Consider the production equations (with w = 0), using R' for R, and p'_a, …, p'_k for p_a, …, p_k. Then multiply the p''s respectively by q"_a, …, q"_k. We obtain

   q"_a (A_ap'_a + B_ap'_b + … + K_ap'_k)(1 + R') = q"_aAp'_a
   q"_b (A_bp'_a + B_bp'_b + … + K_bp'_k)(1 + R') = q"_bBp'_b
   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
   q"_k (A_kp'_a + B_kp'_b + … + K_kp'_k)(1 + R') = q"_kKp'_k
   

3. Add all the equations in our system (from step 2) together:

   [ q"_a (A_ap'_a + B_ap'_b + … + K_ap'_k)
        + q"_b (A_bp'_a + B_bp'_b + … + K_bp'_k)
        + …
        + q"_k (A_kp'_a + B_kp'_b + … + K_kp'_k)](1 + R')
      = (q"_aAp'_a + q"_bBp'_b + … + q"_kKp'_k)
   

4. Now, if we work with the q-equations (as given in §30) taking R" for R and q"_a, …, q"_k for q_a, …, q_k; then multiplying them respectively by p'_a, …, p'_k, we obtain

   p'_a (A_aq"_a + A_bq"_b + … + A_kq"_k)(1 + R") = p'_aAq"_a
   p'_b (B_aq"_a + B_bq"_b + … + B_kq"_k)(1 + R") = p'_bBq"_b
   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
   p'_k (K_aq"_a + K_bq"_b + … + K_kq"_k)(1 + R") = p'_kKq"_k
   

5. We add up all the equations in step 4 to get the equation

   [ p'_a (A_aq"_a + A_bq"_b + … + A_kq"_k)
        + p'_b (B_aq"_a + B_bq"_b + … + B_kq"_k)
        + …
        + p'_k (K_aq"_a + K_bq"_b + … + K_kq"_k)](1 + R")
   = p'_aAq"_a + p'_bBq"_b + … + p'_kKq"_k
   

6. The terms in the equation from step 1 are identical with those of the equation in step 2, despite grouped differently. The only exception? We have R' in one, and R" in the other.

   Thus for the equations to be true, both sides of both equations must be equal to zero: which (since all the p's are positive) implies some of the q"'s must be negative.

   • This prove if there exists a set of positive values for the p's, then there can be no more than one set of positive values for the q's.
   • Footnote. Sraffa notes a similar argument, only putting in the p"'s and the q''s instead of the p''s and the q"'s proves: if there exists a set of positive values for the q's, then there can be no more than one set of positive values for the p's.
7. We had previously seen (in §37) there always exists a set of positive q's and (in §39) there always exists a set of positive p's. We can therefore conclude there exists always one and only one value of R and a corresponding set of positive multipliers (q's) which transform a given economic system into a Standard system.

42. Positive multipliers correspond to lowest value of R

• We can show a corollary: the value of R which corresponds to all-positive prices (which we shall call R') is the lowest of the k possible values of R.
• We assume for contradiction this is not true. Then there exists a value of R lower than R' which we shall call R". As an example, make R' = 15% and R" = 10%.
• To determine if this is possible, we revert to the system with w and r (§11). We assign as wage a quantity of the Standard commodity, which corresponds to R'. Thus we replace the labor terms (L_aw, L_bw, etc.) with proportionate quantities of the Standard commodity, such that their total is a fraction of the Standard national income:

$$1-<!--\; -\; -->\frac{R^{\prime \prime }}{R^{\prime }}$$

• (In the example we have chosen, this would be 1/3).
• At the same time we take as standard of prices an arbitrarily chosen basic commodity a and make its value equal to unity (i.e., p_a = 1).
• Consider two sets of solutions for the resulting system. One corresponds to R' giving us $$r=R^{\prime }\left(1-<!--\; -\; -->\frac{1}{3}\right)=10\%$$

  and all-positive prices (since — being positive at r = R' — they will always be positive for all values of r ≥ 0; c.f. §39).

• The second set of solutions corresponds to R". We know from the last section, when prices correspond to R", the value of the Standard commodity (formed in proportions correspond to R') is zero. So wages vanish and

  r = R" = 10%.
  

  This implies among the prices corresponding to R" some must be negative and others positive.

• The two sets thus give the same value (10%) for r but two different sets of prices.
• But this is impossible: given any single value for r, there exists only one corresponding set of prices. In effect, when r is replaced by a known number (e.g., 10%) the equations form a linear system and for the remaining unknowns there exists a unique set of solutions.
  • Footnote. In these conditions, one of the equations is implicit in the others (see §3, last paragraph) and the number (k - 1) of independent equations is equal to the number of the remaining unknowns.
• Thus R' (the value of R which corresponds to all-positive prices) cannot be higher – and hence must be lower – than any other value R" which corresponds to some positive and some negative prices.
  • Footnote. It may be noted the straight line relation represented by

    r = R (1 - w)

    would continue to hold if wage were measured in any of the other Standard commodities which correspond to the possible values of R higher than R' (if it is possible to conceive of Standard commodities which include negative components; Sraffa addresses this in Ch. 8).

    The prices for various Standard commodities (relative to each other) would with change of r move such that – although wage would represent different proportions of the respective Standard national incomes – these different fractions of different Standard incomes would all be of equal value.

    When r was made equal to R' the wage in terms of any one of the Standard commodities would consist of a nonzero quantity of such Standard commodity…but the value of the latter would be zero if expressed in terms of the Standard commodity formed by means of all-positive multipliers and which corresponds to R'.

43. Standard product replace by equivalent quantity of labor

• The Standard commodity has been a purely auxiliary construction. We can present the essential element of the mechanism without having to resort to the Standard commodity.
• What do we know? If we make the Standard net product equal to unity (i.e., set it to 1), so we measure wage in terms of it, then a relation of proportionality is established between "wage deductions" and "enlarging the rate of profits" (quotes added to indicate the two quantities). Its in accordance with the expression

  r = R'(1 - w),

  where R' is the ratio of the Standard net product to its emans of production, which results from the q-equations.
• The proportion is reversible.
• If we make it a condition of the system that w and r should obey this sort of rule, the wage and commodity prices are then consequently expressed in Standard net product…without need of defining its composition (since no other unit can fulfill the rule)!
• How to do this? We have to substitute for the equation making Standard net product equal to unity (in §34), the relation linking w and r with R'.
  • To find R' (i.e., the value of R corresponding to positive multiplier and prices) we don't have to solve the q-equations. We can find it as the Maximum rate of profits from the previous equations, by making w = 0.
  • Sraffa capitalizes "Maximum" in "Maximum rate of profit" here, though I don't know if it's significant or a typo.
• This condition is sufficient to ensure wage and commodity-prices are expressed in terms of the Standard net product. (Sraffa notes how amazing it is we can use a standard without knowing what it consists of).
• There exists a more tangible measure for prices of commodities, making it possible to displace the Standard net product. The measure is "the quantity of labor which can be purchased by the Standard net product."
  • As soon as we have fixed the rate of profits, without knowing the prices of commodity (nor needing to), a parity is established between the Standard net product and a quantity of labor which depends only on the rate of profits.
  • The resulting prices of commodities can be indifferently regarded as expressed in either (a) the Standard net product, or (b) the quantity of labor — which at the given rate of profits — is known to be equivalent to it.
  • This quantity of labor will inversely vary with the Standard wage (w) and directly with the rate of profits.
  • If the annual labor of this system is taken as unit, this equivalent quantity of labor (derived from the relation above) is $$\frac{1}{w}=\frac{R^{\prime }}{R^{\prime }-<!--\; -\; -->r}.$$
• All the properties for "an invariant standard of value" (as described in §23) are found in the variable quantity of labor, which varies according to a simple rule independent of prices: this unit of measurement increases in magnitude with the fall of the wage, i.e. with the rise of the rate of profits. It varies from (a) equaling the annual labor of the system when the rate of profits vanish, to (b) without limit as the rate of profits approaches its maximum value R'.
• The remaining use of the Standard net product is as the medium which wage is expressed. Sraffa notes in this case "there seems to be no way of replacing it."
  • If we wish to eliminate it altogether, we must cease to regard w as an expression for wage and treat is as a pure number which helps define the quantity of labor which constitutes the unit of prices t the given rate of profits.
  • Then the prices of commodities being expressed in terms of such quantity of labor, we can find its wage in terms of any commodity through taking the reciprocal of the price for that commodity.

44. Wage or rate of profits as independent variable

• The last steps of the preceding argument led us to reverse the practice followed from the outset: treating the wage rater than the rate of profits as the independent variable (or "given" quantity).
• The choice of wage as independent variable was due to its being there as consisting of specified necessaries independent of prices or rate of profits.
• As soon as the possibility of variations in the division of product is admitted, this consideration loses its force.
• When wage is regarded as "given" in terms of a more-or-less abstract standard — and does not acquire definite meaning until prices of commodities are determined — the position is reversed.
  • The rate of profits (as a ratio) has a significance which is independent of any prices, and can well be "given" before prices are fixed.
  • It is accordingly susceptible of being determined from outside the system of production, in particular by the level of money rates of interest.
• The following sections will treat the rate of profits as the independent variable.

Recommended economics writing: Link Exchange

There are a number of interesting things to read (and watch):

1. Course on Game Theory (John Baez)
2. What on Earth is Utility, Anyway? (Unlearning Economics)
3. Britain: A Nation in Decay (Ha-Joon Chang)
4. Victoria Chick: Why don't Economists understand money? (Naked Keynesianism)
5. Empirical Bifurcation Diagram For Kaldor Model (Robert Vienneau)